Problem: Find the least common multiple $(\text{LCM})$ of $8y^6+144y^5+640y^4$ and $2y^4+40y^3+200y^2$. You can give your answer in its factored form.
Solution: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $8y^6+144y^5+640y^4$ can be factored as ${(2)(y^2)}{(2^2)(y^2)}{(y+10)}{(y+8)}$ by factoring out a $8y^4$ and using the sum-product pattern. $2y^4+40y^3+200y^2$ can be factored as ${(2)(y^2)(y+10)}{(y+10)}$ by factoring out a $10y^2$ and using the perfect square pattern. We can see that: Both polynomials share the factors ${(2)(y^2)(y+10)}$ Only the first polynomial has the factors ${(2^2)(y^2)(y+8)}$ Only the second polynomial has the factor ${(y+10)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(2)(y^2)(y+10)}{(2^2)(y^2)(y+8)}{(y+10)}\\\\ &=8(y^4)(y+10)^2(y+8)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $8(y^4)(y+10)^2(y+8)$.